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arXiv:nlin/0610031 · Symmetry Integrability and Geometry Methods and Applications

Integrable Models of Interaction of Matter with Radiation

V. I. Inozemtsev

Published Oct 13, 2006• 2 citations
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Abstract

The simplified models of interaction of charged matter with resonance modes of radiation generalizing the well-known Jaynes-Cummings and Dicke models are considered.It is found that these new models are integrable for arbitrary numbers of dipole sources and resonance modes of the radiation field.The problem of explicit diagonalisation of corresponding Hamiltonians is discussed.

DOI
10.3842/sigma.2006.069

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The simplified models of interaction of charged matter with resonance modes of radiation generalizing the well-known Jaynes-Cummings and Dicke models are considered.It is found that these new models are integrable for arbitrary numbers of dipole sources and resonance modes of the radiation field.The problem of explicit diagonalisation of corresponding Hamiltonians is discussed.
The unitary transformation method is utilized for the closed form solution of the nonlinear generalization of the Jaynes–Cummings (JC) model, which includes arbitrary multiphoton and intensity-dependent interactions between radiation and matter. Specifically, the unitary transformation leading to the diagonalization of the relevant Hamiltonian is established and energy levels and eigenstates are derived. It is also shown that the results for the JC model can be extended to the anti-Jaynes–Cummings (AJC) model through the map between the two models.
The Jaynes-Cummings (JC) model has been studied extensively because of its simple and realistic representation of the dipole interaction between an atom and the quantized radiation field. It exhibits, without perturbative expansions or statistical decorrelations, many quantum mechanical effects in the case of strong or weak interaction, long or short time regime, on or off the resonance. The model is extended to a three-level atom for the investigation of physical phenomena associated with two-photon processes.
Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.
Abstract Radiation damping of an accelerated particle coupled with the field is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena that includes transport properties. In a series of our recent articles we proposed a radically new approach based on the extension of the dynamics of integrable systems. There, we considered a simple model: a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of a charged particle and the actions of the field modes. This makes the system an example of Poincaré nonintegrable systems. We extended the unitary operator to a nonunitary operator Λ. This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can write the Hamilton equations. But we have the opportunity to go to new descriptions. The invertible Λ transformation gives many new aspects, which are hidden in the initial description. For example, we showed that there are fluctuations in the emitted field. In the current article we apply the Λ transformation to describe the evolution of the emitted field from the charged particle and show that the fluctuations lead to a Langevin equation for the field modes. As a consequence, we have a stochastic wave equation that is an extension of Maxwell–Lorentz equation for classical matter–field interacting systems. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
The interaction between matter and fields is a classical problem. Still there are difficulties. Time symmetry is broken. So the radiation damping belongs to a class of phenomena, which includes transport properties. We propose a radically different approach based on the extension of the dynamics of integrable systems. We consider a simple model, a harmonic oscillator interacting with a field. For integrable systems, it is well known that there exists a unitary transformation U. However, in the radiation damping we have resonances between the action of the particle and the actions of the field. This makes the system an example of Poincar\'e nonintegrable systems. We extend the unitary operator to a new star-unitary operator $\ensuremath{\Lambda}.$ This changes the dynamical description of radiation damping. Once we know the Hamiltonian, we can of course write the Hamilton equations. But we have the possibility to go to new descriptions. The invertible $\ensuremath{\Lambda}$ transformation gives many new aspects, which are hidden in the initial description. For example, we show that there are fluctuations, and that there exists an irreducible probability description. The transformation to $\ensuremath{\Lambda}$ representation corresponds to a transformation to Markovian probability equations. We can always come back to the initial representation by the inverse transformation. We have verified this remarkable prediction by detailed numerical calculations. We need the $\ensuremath{\Lambda}$ transformation to obtain the definition of a dressed unstable mode, which has a well-defined lifetime. In the initial representation there are various time scales and there is no strictly exponential lifetime. The situation is the same as the one we have studied in the quantum case in recent papers.